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Perturbative gadgets are used to construct a quantum Hamiltonian whose low-energy subspace approximates a given quantum $k$-body Hamiltonian up to an absolute error $epsilon$. Typically, gadget constructions involve terms with large interaction strengths of order $text{poly}(epsilon^{-1})$. Here we present a 2-body gadget construction and prove that it approximates a target many-body Hamiltonian of interaction strength $gamma = O(1)$ up to absolute error $epsilonllgamma$ using interactions of strength $O(epsilon)$ instead of the usual inverse polynomial in $epsilon$. A key component in our proof is a new condition for the convergence of the perturbation series, allowing our gadget construction to be applied in parallel on multiple many-body terms. We also show how to apply this gadget construction for approximating 3- and $k$-body Hamiltonians. The price we pay for using much weaker interactions is a large overhead in the number of ancillary qubits, and the number of interaction terms per particle, both of which scale as $O(text{poly}(epsilon^{-1}))$. Our strong-from-weak gadgets have their primary application in complexity theory (QMA hardness of restricted Hamiltonians, a generalized area law counterexample, gap amplification), but could also motivate practical implementations with many weak interactions simulating a much stronger quantum many-body interaction.
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Perturbative gadgets are general techniques for reducing many-body spin interactions to two-body ones using perturbation theory. This allows for potential realization of effective many-body interactions using more physically viable two-body ones. In parallel with prior work (arXiv:1311.2555 [quant-ph]), here we consider minimizing the physical resource required for implementing the gadgets initially proposed by Kempe, Kitaev and Regev (arXiv:quant-ph/0406180) and later generalized by Jordan and Farhi (arXiv:0802.1874v4). The main innovation of our result is a set of methods that efficiently compute tight upper bounds to errors in the perturbation theory. We show that in cases where the terms in the target Hamiltonian commute, the bounds produced by our algorithm are sharp for arbitrary order perturbation theory. We provide numerics which show orders of magnitudes improvement over gadget constructions based on trivial upper bounds for the error term in the perturbation series. We also discuss further improvement of our result by adopting the Schrieffer-Wolff formalism of perturbation theory and supplement our observation with numerical results.
We present a systematic method to implement a perturbative Hamiltonian diagonalization based on the time-dependent Schrieffer-Wolff transformation. Applying our method to strong parametric interactions we show how, even in the dispersive regime, full Rabi model physics is essential to describe the dressed spectrum. Our results unveil several qualitatively new results including realization of large energy-level shifts, tunable in magnitude and sign with the frequency and amplitude of the pump mediating the parametric interaction. Crucially Bloch-Siegert shifts, typically thought to be important only in the ultra-strong or deep-strong coupling regimes, can be rendered large even for weak dispersive interactions to realize points of exact cancellation of dressed shifts (`blind spots) at specific pump frequencies. The framework developed here highlights the rich physics accessible with time-dependent interactions and serves to significantly expand the functionalities for control and readout of strongly-interacting quantum systems.
Application of the adiabatic model of quantum computation requires efficient encoding of the solution to computational problems into the lowest eigenstate of a Hamiltonian that supports universal adiabatic quantum computation. Experimental systems are typically limited to restricted forms of 2-body interactions. Therefore, universal adiabatic quantum computation requires a method for approximating quantum many-body Hamiltonians up to arbitrary spectral error using at most 2-body interactions. Hamiltonian gadgets, introduced around a decade ago, offer the only current means to address this requirement. Although the applications of Hamiltonian gadgets have steadily grown since their introduction, little progress has been made in overcoming the limitations of the gadgets themselves. In this experimentally motivated theoretical study, we introduce several gadgets which require significantly more realistic control parameters than similar gadgets in the literature. We employ analytical techniques which result in a reduction of the resource scaling as a function of spectral error for the commonly used subdivision, 3- to 2-body and $k$-body gadgets. Accordingly, our improvements reduce the resource requirements of all proofs and experimental proposals making use of these common gadgets. Next, we numerically optimize these new gadgets to illustrate the tightness of our analytical bounds. Finally, we introduce a new gadget that simulates a $YY$ interaction term using Hamiltonians containing only ${X,Z,XX,ZZ}$ terms. Apart from possible implications in a theoretical context, this work could also be useful for a first experimental implementation of these key building blocks by requiring less control precision without introducing extra ancillary qubits.
In this paper, we generalize the concept of strong quantum nonlocality from two aspects. Firstly in $mathbb{C}^dotimesmathbb{C}^dotimesmathbb{C}^d$ quantum system, we present a construction of strongly nonlocal quantum states containing $6(d-1)^2$ orthogonal product states, which is one order of magnitude less than the number of basis states $d^3$. Secondly, we give the explicit form of strongly nonlocal orthogonal product basis in $mathbb{C}^3otimes mathbb{C}^3otimes mathbb{C}^3otimes mathbb{C}^3$ quantum system, where four is the largest known number of subsystems in which there exists strong quantum nonlocality up to now. Both the two results positively answer the open problems in [Halder, textit{et al.}, PRL, 122, 040403 (2019)], that is, there do exist and even smaller number of quantum states can demonstrate strong quantum nonlocality without entanglement.
We study the perturbative unitarity bound given by dimension six derivative interactions consisting of Higgs doublets. These operators emerge from kinetic terms of composite Higgs models or integrating out heavy particles that interact with Higgs doublets. They lead to new phenomena beyond the Standard Model. One of characteristic contributions by derivative interactions appear in vector boson scattering processes. Longitudinal modes of massive vector bosons can be regarded as Nambu Goldstone bosons eaten by each vector field with the equivalence theorem. Since their effects become larger and larger as the collision energy of vector bosons increases, vector boson scattering processes become important in a high energy region around the TeV scale. On the other hand, in such a high energy region, we have to take the unitarity of amplitudes into account. We have obtained the unitarity condition in terms of the parameter included in the effective Lagrangian for one Higgs doublet models. Applying it to some of models, we have found that contributions of derivative interactions are not so large enough to clearly discriminate them from the Standard Model ones. We also study it in two Higgs doublet models. Because they are too complex to obtain the bound in the general effective Lagrangian, we have calculated it in explicit models. These analyses tell us highly model dependence of the perturbative unitarity bounds.
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